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Theorem qsqeqor 10616
Description: The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.)
Assertion
Ref Expression
qsqeqor ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))

Proof of Theorem qsqeqor
StepHypRef Expression
1 qre 9614 . . . . . . 7 (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
21ad3antrrr 492 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐴 ∈ ℝ)
3 simplr 528 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐴)
4 qre 9614 . . . . . . 7 (𝐵 ∈ ℚ → 𝐵 ∈ ℝ)
54ad3antlr 493 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
6 simpr 110 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
7 sq11 10578 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
82, 3, 5, 6, 7syl22anc 1239 . . . . 5 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
9 orc 712 . . . . 5 (𝐴 = 𝐵 → (𝐴 = 𝐵𝐴 = -𝐵))
108, 9syl6bi 163 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵𝐴 = -𝐵)))
11 oveq1 5876 . . . . . . 7 (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2))
1211a1i 9 . . . . . 6 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2)))
13 oveq1 5876 . . . . . . . . 9 (𝐴 = -𝐵 → (𝐴↑2) = (-𝐵↑2))
1413adantl 277 . . . . . . . 8 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (-𝐵↑2))
15 qcn 9623 . . . . . . . . . 10 (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
16 sqneg 10565 . . . . . . . . . 10 (𝐵 ∈ ℂ → (-𝐵↑2) = (𝐵↑2))
1715, 16syl 14 . . . . . . . . 9 (𝐵 ∈ ℚ → (-𝐵↑2) = (𝐵↑2))
1817ad2antlr 489 . . . . . . . 8 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (-𝐵↑2) = (𝐵↑2))
1914, 18eqtrd 2210 . . . . . . 7 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2))
2019ex 115 . . . . . 6 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = -𝐵 → (𝐴↑2) = (𝐵↑2)))
2112, 20jaod 717 . . . . 5 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴 = 𝐵𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
2221ad2antrr 488 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
2310, 22impbid 129 . . 3 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
2417eqeq2d 2189 . . . . . 6 (𝐵 ∈ ℚ → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
2524ad3antlr 493 . . . . 5 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
261ad3antrrr 492 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℝ)
27 simplr 528 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ 𝐴)
28 qnegcl 9625 . . . . . . . . 9 (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
29 qre 9614 . . . . . . . . 9 (-𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
3028, 29syl 14 . . . . . . . 8 (𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
3130ad3antlr 493 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
32 simpr 110 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
334le0neg1d 8464 . . . . . . . . 9 (𝐵 ∈ ℚ → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
3433ad3antlr 493 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
3532, 34mpbid 147 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
36 sq11 10578 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
3726, 27, 31, 35, 36syl22anc 1239 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
38 olc 711 . . . . . 6 (𝐴 = -𝐵 → (𝐴 = 𝐵𝐴 = -𝐵))
3937, 38syl6bi 163 . . . . 5 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) → (𝐴 = 𝐵𝐴 = -𝐵)))
4025, 39sylbird 170 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵𝐴 = -𝐵)))
4121ad2antrr 488 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
4240, 41impbid 129 . . 3 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
43 0z 9253 . . . . . 6 0 ∈ ℤ
44 zq 9615 . . . . . 6 (0 ∈ ℤ → 0 ∈ ℚ)
4543, 44ax-mp 5 . . . . 5 0 ∈ ℚ
46 qletric 10230 . . . . 5 ((0 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐵𝐵 ≤ 0))
4745, 46mpan 424 . . . 4 (𝐵 ∈ ℚ → (0 ≤ 𝐵𝐵 ≤ 0))
4847ad2antlr 489 . . 3 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → (0 ≤ 𝐵𝐵 ≤ 0))
4923, 42, 48mpjaodan 798 . 2 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
50 qnegcl 9625 . . . . . . . . . 10 (𝐴 ∈ ℚ → -𝐴 ∈ ℚ)
51 qre 9614 . . . . . . . . . 10 (-𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
5250, 51syl 14 . . . . . . . . 9 (𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
5352ad3antrrr 492 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → -𝐴 ∈ ℝ)
54 simplr 528 . . . . . . . . 9 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐴 ≤ 0)
551le0neg1d 8464 . . . . . . . . . 10 (𝐴 ∈ ℚ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
5655ad3antrrr 492 . . . . . . . . 9 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
5754, 56mpbid 147 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ -𝐴)
584ad3antlr 493 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
59 simpr 110 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
60 sq11 10578 . . . . . . . 8 (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
6153, 57, 58, 59, 60syl22anc 1239 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
6261biimpd 144 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) → -𝐴 = 𝐵))
63 qcn 9623 . . . . . . . . . 10 (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
64 sqneg 10565 . . . . . . . . . 10 (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))
6563, 64syl 14 . . . . . . . . 9 (𝐴 ∈ ℚ → (-𝐴↑2) = (𝐴↑2))
6665adantr 276 . . . . . . . 8 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴↑2) = (𝐴↑2))
6766eqeq1d 2186 . . . . . . 7 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
6867ad2antrr 488 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
69 negcon1 8199 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
7063, 15, 69syl2an 289 . . . . . . . 8 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
71 eqcom 2179 . . . . . . . 8 (-𝐵 = 𝐴𝐴 = -𝐵)
7270, 71bitrdi 196 . . . . . . 7 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵𝐴 = -𝐵))
7372ad2antrr 488 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (-𝐴 = 𝐵𝐴 = -𝐵))
7462, 68, 733imtr3d 202 . . . . 5 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → 𝐴 = -𝐵))
7574, 38syl6 33 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵𝐴 = -𝐵)))
7621ad2antrr 488 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
7775, 76impbid 129 . . 3 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
7852ad3antrrr 492 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐴 ∈ ℝ)
79 simplr 528 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ≤ 0)
8055ad3antrrr 492 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
8179, 80mpbid 147 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐴)
8230ad3antlr 493 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
83 simpr 110 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
8433ad3antlr 493 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
8583, 84mpbid 147 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
86 sq11 10578 . . . . . . 7 (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
8778, 81, 82, 85, 86syl22anc 1239 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
8865, 17eqeqan12d 2193 . . . . . . 7 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
8988ad2antrr 488 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
9063ad3antrrr 492 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℂ)
9115ad3antlr 493 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ∈ ℂ)
9290, 91neg11ad 8254 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (-𝐴 = -𝐵𝐴 = 𝐵))
9387, 89, 923bitr3d 218 . . . . 5 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
9493, 9syl6bi 163 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵𝐴 = -𝐵)))
9521ad2antrr 488 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
9694, 95impbid 129 . . 3 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
9747ad2antlr 489 . . 3 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → (0 ≤ 𝐵𝐵 ≤ 0))
9877, 96, 97mpjaodan 798 . 2 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
99 qletric 10230 . . . 4 ((0 ∈ ℚ ∧ 𝐴 ∈ ℚ) → (0 ≤ 𝐴𝐴 ≤ 0))
10045, 99mpan 424 . . 3 (𝐴 ∈ ℚ → (0 ≤ 𝐴𝐴 ≤ 0))
101100adantr 276 . 2 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐴𝐴 ≤ 0))
10249, 98, 101mpjaodan 798 1 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wcel 2148   class class class wbr 4000  (class class class)co 5869  cc 7800  cr 7801  0cc0 7802  cle 7983  -cneg 8119  2c2 8959  cz 9242  cq 9608  cexp 10505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-seqfrec 10432  df-exp 10506
This theorem is referenced by:  4sqlem10  12368
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